Many techniques are taught in advanced solid state courses but they are almost all derived for perfectly crystalline materials. For example, band structure really only appears theoretically when you look at periodic potentials that are pretty big in at least one direction.

But in experiment you often end up using materials that are polycrystalline (e.g., an evaporated film). Then, it is only periodic inasmuch the individual grains making up the sample are periodic, but they are both small (if your grains are ~1um wide, then you really only have 1000-10000 atoms in one direction, per grain) and randomly oriented, which messes up any theory I possibly know.

How can one analyze these? Is it possible? Which theories can still be used reasonably and which have to be thrown out?

Thank you!

edit: Sorry, I realized this could be a little ambiguous. I know you can probably model them computationally, but I mean more analytically, not just brute force.

$\begingroup$Well, depending on the characteristic length scales of the system, 1000-10000 atoms might be more than enough to ensure that each grain contains "periodic" physics in the bulk. I can't say much about about the edge states though. Additionally, x-ray diffraction on powder samples is an extremely common experiment and I think the results are only marginally affected by the lack of bulk crystal (aka the line-widths increase but not much else).$\endgroup$
– Todd RMar 31 '15 at 21:02

1 Answer
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I can't give a general answer, but let me give an example. In graphene, grain boundaries have been modeled as dislocations. Their effect on transport properties can be estimated from an (analytic) Dirac equation model, and these estimates can be verified computationally, in this case using nonequilibrium Green's functions methods.